Let be B(z) the exponential generating function for the number $b_n$ of different rooted unordered binary trees with exactly n leaves labeled only at their leaves (so the internal nodes are unlabeled). Then it's a well know result that $b_n = 1\cdot3\cdot\ldots\cdot(2n-3)=(2n-3)!!$ (see eg. Schröder "Vier combinatorische Probleme"). The EGF satisfies the recursion

$B(z) = z + \frac{1}{2}B(z)^2$.

There is one tree with one leave and every tree is built up from two (smaller) trees where the order doesn't matter (because the trees are unplane).